Finite volume simulation of a semi-linear Neumann problem (Keller-Segel model) on rectangular domains

TL;DR

This paper implements a finite volume method to numerically solve a semilinear Keller-Segel model with Neumann boundary conditions, demonstrating various solution shapes in 3D and contour plots based on parameters.

Contribution

It introduces a finite volume discretization approach for the Keller-Segel model and provides numerical solutions illustrating different pattern formations.

Findings

Various single and multi-peaked solution shapes identified

Numerical solutions visualized in 3D and contour plots

Parameter effects on solution patterns analyzed

Abstract

In this study, the finite volume method is implemented for solving the problem of the semilinear equation: $-d \delta u+ u=u^q (d, q>0) $with a homogeneous Neumann boundary condition. This problem is equivalent to the known stationary Keller-Segel model, which arises in chemotaxis.After discretization, a nonlinear algebraic system is obtained and solved on the platform Matlab. As a result, many single peaked and multi-peaked shapes in 3D and contour plots can be drawn depending on the parameters d and q.